Tolerance Panel

An iterative approach is used to obtain pipe flow solutions. When approaching any analysis iteratively, a stopping point must be defined. The simplest approach is to stop after a certain number of iterations. However, this either results in uncertain results because the result is still changing, or wasted processing time because there are too many iterations.

A solution is to use iterative tolerance. This tolerance is related to the change between iterations. When this change is smaller than the iterative tolerance, the solution is said to be converged. These settings are specified in the Tolerance panel.

Tolerance and Solution Accuracy

Iterative tolerance does not have any relationship to an engineering tolerance.

An engineering tolerance is a familiar term to most engineers - it is a permissible limit in variation of a parameter. A simple example is the length of a steel bar - an engineer might specify that this length must be 3 ± 0.01 meters. A steel bar manufactured by this specification is known to be within the engineering tolerance of the "true" or desired length. In other words, engineering tolerance describes an accuracy.

Iterative tolerances do not directly describe an accuracy. Convergence only means that the change between iterations was within the specified limit.

With sound iterative methods, and an appropriately specified iterative tolerance, the iterative solution will be close to the true solution. However, even with good settings, the iterative tolerance does not define how close the results are to the true solution.

If there is concern that the results may not be close enough to the true answer, the tolerances could be lowered and the results compared. If the results do not change appreciably, an appropriate tolerance was likely selected.

Avoiding False Convergence

It is important to keep the tolerance at least one order of magnitude smaller (two orders or more is recommended) than the relaxation. A false convergence can occur when the change from the old value to the new value is small enough to be within the tolerance because it is restricted by the relaxation. False convergence in the steady state can lead to artificial transients in the Transient Solver.

Absolute Tolerance

The absolute change method is generally more reliable for iterative solvers like the steady state solver for AFT Impulse because it is less sensitive to the magnitude of the solution, whether close to zero or very large. However, in principle, specifying absolute tolerance requires some knowledge of the final solution. If you specify mass flow convergence as sufficient when the flow no longer changes by 1 lbm/sec each iteration, then your solution will be compromised if the magnitude is also near 1 lbm/sec. In this case it would be better to set the tolerance 3 or 4 orders of magnitude below the solution.

Relative Tolerance

Relative tolerances, on the other hand, only check the relative change of the number. That is, each successive change is divided by the number itself. This removes the requirement to be knowledgeable about the steady state solution, but can give trouble for problems where one of the solutions is too close to zero (because the change is divided by a number close to zero, making a very large number).

There are options for combining absolute and relative tolerance. You can tell AFT Impulse to assess convergence based on whether either an absolute or relative criteria is satisfied, or whether both absolute and relative criteria are satisfied.

Table 1 shows an example of how the four different criteria would be applied to a flow rate iterations.

Table 1: Example of four tolerance methods for a mass flow rate example. Absolute tolerance = 0.0005 lbm/s, Relative tolerance = 0.0001